Short answers: no, they are not the same; they are somewhat related. A more detailed discussion follows.
Indeed, in most QFT books zero temperature is usually assumed. However, if one is interested in energy scales that are way beyond the temperature of the system, the zero-temperature approximation is a valid one. For example, the thermal energy at room temperature is around 25 meV, which means that optical photons are not being thermally created (their energy in vacuum is 1 eV $\ll$ 25 meV), not even talking about thermal generation of electrons (0.5 MeV). Therefore, assuming zero temperature is a very good approximation for, e.g., QED in particle accelerators.
Furthermore, the grand unification scale sets such a high effective temperature that all the physics we know occurs in the effectively zero-temperature limit (with the exception of the early universe, as you have pointed out).
In general, an effective theory at a certain energy scale using a renormalization scheme (renormalization group, RG) is obtained by integrating out all the degrees of freedom higher than that energy scale. In a QFT, the dynamics of these degrees of freedom are determined by, for example, quantum fluctuations and inter-particle interactions. However, a thermal field theory has to do with statistical mechanics, and thus uses the language of ensembles and thermal baths. Moreover, it is possible to perform renormalization procedures for thermal systems as well. In that case, in addition to quantum fluctuations, there are thermal ones. Therefore, the resulting RG equations in general depend on the temperature.
Unfortunately, I do not know of a finite-temperature RG example in vacuum (if that is even well defined). However, there is plenty of examples of RG equations in some medium. One example can be found in Ultracold Quantum Fields by Stoof et al., where in the RG-flow equations (14.64) - (14.66) there are Fermi-distribution functions $N_\uparrow$ that explicitly depend on temperature.